{
  "abstract": "At the critical Hardy Gaussian weight for the Schrödinger equation in one space dimension on $[0,1]$, the known nonzero scalar example in the weighted $L^2$ class carries a complex-valued potential. Cassano and Fanelli observed that the existence of a real-valued scalar endpoint example was open, and produced examples with real electric and magnetic potentials only after introducing a magnetic potential. We construct a nonzero smooth solution of $i\\partial_t u+\\partial_x^2 u=Vu$ on $\\mathbb{R}\\times[0,1]$ such that $e^{x^2/4}u(\\cdot,0),e^{x^2/4}u(\\cdot,1)\\in L^2(\\mathbb{R})$ and the potential is bounded, smooth, real-valued, and purely scalar and is supported in one fixed compact spatial interval for all times. This compact-support endpoint example is the main result. We also record the explicit rational-tail realization underlying the construction. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20202",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiao-Ming Fu"
    },
    {
      "name": "Tianyang Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/983kqv392r99a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20202",
  "fulltext_url": null,
  "id": "983kqv392r99a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20202",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.20202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/983kqv392r99a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions",
  "title_text": "Compactly supported real scalar potentials realizing the Hardy uncertainty endpoint for Schrödinger evolutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/983kqv392r99a",
  "venue": "arXiv",
  "versions": []
}